nLab
representable morphism of stacks
Context
$(\infty,1)$Topos Theory
(∞,1)topos theory
Background
Definitions

elementary (∞,1)topos

(∞,1)site

reflective sub(∞,1)category

(∞,1)category of (∞,1)sheaves

(∞,1)topos

(n,1)topos, ntopos

(∞,1)quasitopos

(∞,2)topos

(∞,n)topos
Characterization
Morphisms
Extra stuff, structure and property

hypercomplete (∞,1)topos

over(∞,1)topos

nlocalic (∞,1)topos

locally nconnected (n,1)topos

structured (∞,1)topos

locally ∞connected (∞,1)topos, ∞connected (∞,1)topos

local (∞,1)topos

cohesive (∞,1)topos
Models
Constructions
structures in a cohesive (∞,1)topos
Contents
Definition
A morphism $f : X \to Y$ of stacks over a site $C$ is called representable if for all representable objects $U \in C \stackrel{Y}{\hookrightarrow} Stacks(C)$ and all morphisms $U \to Y$ the homotopy pullback $X \times_Y U$ in
$\array{
X \times_Y U &\to& X
\\
\downarrow &{}^{\simeq}\swArrow& \downarrow^f
\\
U &\to& Y
}$
is again representable.
Properties
Pushforward in generalized cohomology
Along representable morphisms $f$ of stacks over smooth manifolds (smooth infinitygroupoids) is induced a pushforward in generalized cohomology operation.
References
The general definition appears for instance as def. 38.5 in
(there with stacks perceived equivalently and dually under the Grothendieck construction as fibered categories).
Applications of pushforward in generalized cohomology along representable morphisms appear for instance in
Revised on November 7, 2012 21:53:48
by
Urs Schreiber
(82.169.65.155)